Experimental and numerical optimization of coal breakage performance parameters through abrasive gas jet

2.1 Nozzle parameter

This paper mainly studies the influence of abrasive gas jet pressure and jet target distance on erosion effect. For the abrasive gas jet, the energy obtained by abrasive from the gas plays a decisive role in jet effect. The common nozzles include convergent nozzle and Laval nozzle. Because of the compressibility of the gas and the lack of divergent segment, the velocity reaches to maximum at the nozzle outlet and forms shock wave. So, the jet flow filed structure isn’t optimal, the abrasive can’t accelerate sufficiently either. The maximum velocity at the outlet of the convergent nozzle can only be the sound velocity, namely, subsonic jet. In contrast, the Laval nozzle can obtain supersonic gas jet. The length and angel of divergent segment control the expansive degree of gas jet and form optimal flow field structure. Therefore, the Laval nozzle is selected to study the parameters of high-pressure abrasive gas jet.

A Laval nozzle is composed of three parts of inlet stability section, convergent section, and the divergent segment. Steady and uniform gas flow can be obtained in the inlet stability section. The convergent section accelerates the gas flow to sound velocity. Finally, supersonic gas flow is obtained in the divergent segment [20]. The size of the Laval nozzle is designed and calculated according to the gas dynamics principle [21-22]. The mass flow of any section of the nozzle is:

$m=\frac{P_{0} A}{\sqrt{R T_{0}}} \sqrt{\frac{2 k_{0}}{k_{0}-1}\left[\left(\frac{p}{P_{0}}\right)^{\frac{2}{k_{0}}}-\left(\frac{p}{P_{0}}\right)^{\frac{k_{0}+1}{k_{0}}}\right]}$   (1)

The area ratio of different sections of the nozzle is:

$\frac{A}{A_{0}}=\frac{1}{M_{a}}\left(\frac{2}{k_{0}+1}\right)^{\frac{k_{0}+1}{2\left(k_{0}-1\right)}}\left(1+\frac{k_{0}-1}{2} M_{a}^{2}\right)^{\frac{k_{0}+1}{2\left(k_{0}-1\right)}}$        (2)

The outlet velocity of the nozzle is:

$v=\sqrt{\frac{2 k_{0}}{k_{0}-1} R T_{0}\left[1-\left(\frac{P}{P_{0}}\right)^{\frac{k_{0}-1}{k_{0}}}\right]}$     (3)

The outlet temperature of the nozzle is:

$T_{2}=T_{0}\left(\frac{P}{P_{0}}\right)^{\frac{k_{0}-1}{k_{0}}}$      (4)

Mach number:

$M_{a}=\frac{v}{\sqrt{k_{0} R T}}$      (5)

where, T₀ is stagnation temperature (K). P₀ is stagnation pressure (Pa). k₀ is adiabatic index, k₀=1.4；R is gas constant. m is mass flow (kg/s). A and A₀ are areas of the sections. Ma is the Mach number. T₂ is the outlet temperature of the nozzle.

The convergent angle and divergent angle of the nozzle are valued according to the experience [23-24]. Generally speaking, the convergent angle is 30°, and the divergent angle is 10°. When the temperature is 300K and the pressure is 15 MPa, the measured mass flow of the inlet gas is 0.096kg/s. Because of the characteristics of the Laval nozzle, the Mach number at the nozzle throat is Ma=1. According to the above formula, the diameter of the nozzle inlet is d=6mm, the diameter of the nozzle throat is d₀=2mm, and the diameter of the nozzle outlet is D=8mm. The length of the convergent section is L₂=7.4mm, and the length of the divergence segment is L₃=34.3mm. The length of the inlet stability section is L₁=6mm. Therefore, the nozzle structure is as shown in Figure 1.

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Figure 1. The structure of Laval nozzle

2.2 Numerical simulation model

Considering the symmetry of the Laval nozzle structure, the two-dimensional model can be utilized for the calculation in the numerical simulation. That is, half of the actual flow area is utilized as the calculation model to divide the grid. The calculation area mainly includes the nozzle and free jet region. According to the actual size of nozzle, the flow field model of nozzle and the free flow field at the nozzle outlet were established by the software GAMBIT. The grid is divided through the quadrilateral structured grid. Considering the difference of target distance, this paper only gives a grid model, as shown in Figure 2, where AB is symmetric boundary, AC is the pressure inlet boundary, CDE is non-slip adiabatic wall, EFG is the pressure outlet boundary, and BG is the wall of the jet erosion, which can reflect the erosion effect of abrasive gas jet. It is a kind of boundary condition which is non-slip boundary condition.

To test the mesh independence, the outlet velocities of gas jet have been compared to the condition of different mesh number, such as 16289, 36850, and 65,729. The results are as follows.

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Figure 2. The mesh generation of nozzle and flow field

Table 1. The mesh independency study

The results of grid-independent verification show that it is reliable to use 36,850 mesh structures to calculate.

In this experiment, the 120-mesh quartz sand abrasive was adopted, and the Moh's hardness was 7. In order to compare with the experimental results, the numerical simulation adopted the 120-mesh quartz sand abrasive. The density was 2660kg/m³, diameter is 0.125mm, and the mass flow of the abrasive was 0.01kg/s. The initial velocity of the abrasive was 0. The outlet pressure was barometric pressure, 101.325kpa. The parameter values of the numerical simulation are shown in Table 2.

Table 2. Parameter values of the numerical simulation

2.3 Calculation model

Numerical simulation adopted the DPM discrete term model in Fluent to simulate the movement erosion of abrasive particles. Firstly, the gas flow field was calculated. When the gas flow field converged, the discrete term was added. This paper adopted RNG k-ε turbulence model, and the N-S equation was solved through the finite volume method. Compared with k-ε model, the accuracy of the simulation is higher. The fluid is ideal gas, and the RNG k-ε turbulence model is described as follows:

$\frac{\partial(\rho k)}{\partial t}+\frac{\partial\left(\rho k u_{i}\right)}{\partial x_{i}}$$=\frac{\partial}{\partial x_{j}}\left[\alpha_{k} \mu_{e f f} \frac{\partial k}{\partial x_{j}}\right]+G_{k}+G_{b}-\rho \varepsilon-Y_{M}$     (6)

$\frac{\partial(\rho \varepsilon)}{\partial t}+\frac{\partial\left(\rho \varepsilon u_{i}\right)}{\partial x_{i}}=\frac{\partial}{\partial x_{j}}\left[\alpha_{\varepsilon} \mu_{e f f} \frac{\partial \varepsilon}{\partial x_{j}}\right]$$+C_{1 s}^{*} \frac{\varepsilon}{k}\left(G_{k}+C_{3 \varepsilon} G_{b}\right)-C_{2 \varepsilon} \rho \frac{\varepsilon^{2}}{k}-R$     (7)

where,

$C_{1 \varepsilon}^{*}=C_{1 \varepsilon}-\frac{\eta\left(1-\eta / \eta_{0}\right)}{1+\beta \eta^{3}}$

$\eta=\left(\alpha E_{i j} \cdot E_{i j}\right)^{1 / 2} \frac{k}{\varepsilon}$

$\mu_{e f f}=\mu+\mu_{t}$

$\mu_{t}=\rho C_{u} \frac{k^{2}}{\varepsilon}$

$G_{b}=\beta g_{i} \frac{\mu_{t}}{\operatorname{Pr}_{t}} \frac{\partial T}{\partial x_{i}}$

$G_{k}=\mu_{t}\left(\frac{\partial \mu_{i}}{\partial x_{j}}+\frac{\partial \mu_{j}}{\partial x_{i}}\right) \frac{\partial \mu_{i}}{\partial x_{j}}$

$Y_{M}=2 \rho \varepsilon \frac{k}{a^{2}}$

where, G_k is the production item of turbulent kinetic energy k caused by the mean velocity gradient. G_b is the production item of turbulent kinetic energy caused by buoyancy. Y_M is the influence of compressible turbulent fluctuation on the total dissipation rate. α_k and α_ε are the reciprocals of effective Prandtl numbers of turbulent kinetic energy and dissipation rate, respectively. Pr_t is turbulent Prandtl number. C_1ε, C_2ε and C_3ε are empirical constants. g_i is the component of gravitational acceleration in the i direction. is the coefficient of thermal expansion, and a is the sound velocity.

2.4 Numerical simulation results

According to the velocity formula of nozzle outlet, the outlet velocity of nozzle under different jet pressure could be calculated theoretically, as shown in Figure 3. It can be observed that the velocity of the outlet gas increased with the inlet pressure, and the increased speed gradually slowed down. When the inlet pressure was greater than 15MPa, the increase of gas velocity became increasingly smaller. Therefore, the jet velocity was increasingly less affected by simply enhancing the jet pressure. Under certain conditions, the optimal pressure exists for the abrasive jet.

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Figure 3. Relationship between gas jet velocity of nozzle outlet and jet pressure

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Figure 4. Relationship between abrasive velocity and jet pressure

According to the abrasive velocity change curve under the different inlet pressure as shown in Figure 4, with the increase of the inlet pressure, the abrasive velocity first increased rapidly. When the pressure was greater than 10MPa, the increased of abrasive velocity slowed down. When the pressure was between 20MPa and 30MPa, the abrasive velocity was relatively stable. The power of abrasive particles of high-pressure abrasive gas jet was from the high-pressure gas. According to the change of gas velocity as shown in Figure 4, when the inlet pressure was less than 10MPa, with the increase of pressure, the gas velocity at the inlet of the nozzle increased substantially. When the inlet pressure was greater than 10MPa and smaller than 20MPa, with the increase of pressure, the increase of the gas phase velocity became slow. When the inlet pressure was greater than 20MPa, the increase of gas velocity was slight. The change of the velocity of the abrasive particle of the abrasive gas jet was consistent with the change trend of gas velocity with pressure.

Fluent was adopted to carry out numerical simulation. The erosion rate module can directly reflect the erosion rate of the target [25], namely, the quality of the material removed per unit area per unit time. In Fluent, the erosion rate was defined as:

$R_{\text {erosion}}=\sum_{p=1}^{N \text { particle }} \frac{m_{p} C\left(d_{p}\right) f(\alpha) v_{p}^{b(v)}}{A_{\text { face }}}$     (8)

where, C(d_p) is particle size function; α is the impact angle between the particle path and the wall surface. f(α) is the impact angle function. v is the relative velocity of particles. b(v) is the relative velocity function of particles. A_face is the area of the wall.

Table 3. The erosion rate of target

In this paper, erosion rate obtained by erosion model is the erosion rate at the same time. According to the simulation results, it is found that the erosion rate is approximately parabolic with the change of the jet pressure. When the pressure was 15MPa, the erosion rate was the largest. When the jet pressure increased, the abrasive velocity was higher. When the abrasive impacted the target, the velocity of the abrasive rebound was larger, which further affected the incoming abrasive and the jet flow effect. Therefore, within a certain range of jet pressure, the jet erosion effect was better when the injection pressure was 15MPa.

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Figure 5. Gas jet velocity under different target distances

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Figure 6. Relationship between target distance and abrasive velocity

Figure 5 shows the change curve of gas velocity with the target distance when the inlet pressure was 15MPa. It can be observed that the velocity of gas phase was fluctuant when the target distance was between 0 and 12cm. In the case that the target distance was greater than 12cm, the gas velocity began to obviously decrease. As shown in Figure 6, when the pressure was constant, the velocity of abrasive particles increased with the target distance. This indicates that the kinetic energy of the gas was converted into the kinetic energy of the abrasive in the acceleration process of gas expansion at the nozzle outlet. When the target distance was greater than 13cm, the abrasive velocity was basically unchanged. At this moment, the gas velocity began to decrease. The acceleration of the abrasive was limited, and the velocity increased gently. In the movement process of abrasive particles, the abrasive velocity was always smaller than the gas velocity.

The erosion effect of abrasive gas jet depends on the energy obtained by abrasive particles. Therefore, a better jet erosion effect can be obtained by choosing appropriate jet pressure and target distance. According to the results of numerical simulation, it can be found that the optimal jet pressure is 15MPa.

(1) A novel method of coal and rock breakage with abrasive gas jet has been proposed. The method is more efficient in coal and rock breakage compared to the gas jet which forms small erosion pit and crack on the coal or rock surface merely.  

(2) The velocity filed and erosion effect of abrasive gas jet have been studied by Fluent, and it can be concluded that the optical pressure of coal breakage is 15MPa.

(3) The relationship between gas jet pressure and erosion depth have been studied via experiment of coal and rock breakage with abrasive gas jet flow. With the increase of the gas jet pressure, the erosion depth and erosion volume begin to increase rapidly, and then tend to be steady. So, there is optimal gas jet pressure for coal or rock breakage. When the gas jet pressure is 15MPa, the coal or rock breakage effect is Optimal.

(4) It can be concluded by experiment that the erosion depth first increases and then decreases with the increase of the target distance when gas jet pressure is invariable. Also, the erosion volume continuously increased. The relationship between erosion depth and target distance is obtained by fitting the experimental data, H=0.0028h3-0.1654h2+2.5468h+12.098. Also, the relationship between the erosion volume and the target distance is obtained, V=-0.0005h3+0.0094h2+0.3553h+0.5509. According to the formula, it can be concluded that the optimal target distance of the coal breakage by abrasive gas jet is 10cm, the maximum target distance of the coal breakage by abrasive gas jet is up to 25cm.